Measures of Central Tendency

What is Central Tendency?

The Three Main Measures

Arithmetic Mean (x̄)

Sum of all values divided by count. The most common measure.

Properties:

• Uses every data point — sensitive to outliers
• The only measure where Σ(xᵢ - x̄) = 0
• Minimizes the sum of squared deviations

Median (M)

The middle value when data is sorted. For even n, average the two middle values.

Properties:

• Robust to outliers
• Appropriate for ordinal and continuous data
• Minimizes the sum of absolute deviations

Mode

The most frequently occurring value(s). A distribution can be unimodal, bimodal, or multimodal.

Properties:

• Only measure appropriate for nominal data
• Can have multiple modes (or none in continuous data)
• Not affected by extreme values

import numpy as np
import pandas as pd
from scipy import stats
import matplotlib.pyplot as plt

def describe_central(data, label):
    mean = np.mean(data)
    median = np.median(data)
    mode_result = stats.mode(data, keepdims=True)
    mode = mode_result.mode[0]
    
    print(f"\n{label}:")
    print(f"  Mean   = {mean:.2f}")
    print(f"  Median = {median:.2f}")
    print(f"  Mode   = {mode:.2f}")
    print(f"  Mean - Median = {mean - median:.2f} {'(right-skewed)' if mean > median else '(left-skewed)' if mean < median else '(symmetric)'}")

np.random.seed(42)

# Symmetric distribution
sym = np.random.normal(50, 10, 1000)
describe_central(sym, "Symmetric (Normal)")

# Right-skewed (income-like)
right_skew = np.random.lognormal(3.5, 0.8, 1000)
describe_central(right_skew, "Right-Skewed (Income)")

# Left-skewed
left_skew = 100 - np.random.exponential(10, 1000)
describe_central(left_skew, "Left-Skewed")

# With outlier
with_outlier = np.concatenate([np.random.normal(50, 5, 99), [500]])
describe_central(with_outlier, "Data with One Extreme Outlier (n=100)")

Effect of Skewness on Mean vs Median

Right-Skewed:        Mode < Median < Mean
Symmetric:           Mode ≈ Median ≈ Mean
Left-Skewed:         Mean < Median < Mode

fig, axes = plt.subplots(1, 3, figsize=(15, 5))

datasets = {
    'Right-Skewed': np.random.lognormal(3, 0.8, 2000),
    'Symmetric': np.random.normal(50, 10, 2000),
    'Left-Skewed': 100 - np.random.exponential(10, 2000)
}

for ax, (title, data) in zip(axes, datasets.items()):
    ax.hist(data, bins=40, density=True, color='lightblue', edgecolor='gray', alpha=0.7)
    mean_val = np.mean(data)
    median_val = np.median(data)
    ax.axvline(mean_val, color='red', lw=2, linestyle='--', label=f'Mean={mean_val:.1f}')
    ax.axvline(median_val, color='blue', lw=2, linestyle='-', label=f'Median={median_val:.1f}')
    ax.set_title(title)
    ax.legend()

plt.tight_layout()
plt.savefig('central_tendency.png', dpi=150)
plt.show()

When to Use Which Measure

A Famous Example: Income

# Simulate US-like income distribution
np.random.seed(0)
incomes = np.concatenate([
    np.random.lognormal(10.8, 0.7, 9990),   # Most people
    np.random.lognormal(13.5, 1.0, 10)      # Very wealthy (top 0.1%)
])

print(f"Mean income:   ${np.mean(incomes):>12,.0f}")
print(f"Median income: ${np.median(incomes):>12,.0f}")
print(f"The top 0.1% pulls the mean ${np.mean(incomes)-np.median(incomes):,.0f} above the median!")

Central Tendency in Machine Learning

Central tendency is fundamental to ML:

import numpy as np
from sklearn.linear_model import LinearRegression, HuberRegressor
from sklearn.metrics import mean_squared_error, mean_absolute_error

np.random.seed(42)

# Generate data with outliers
n = 200
X = np.random.randn(n, 1) * 10
y = 3 * X[:,0] + np.random.randn(n) * 5

# Add outliers
outlier_idx = np.random.choice(n, 10, replace=False)
y[outlier_idx] += np.random.randn(10) * 50

# MSE minimizes to the mean → sensitive to outliers
model_mse = LinearRegression().fit(X, y)
pred_mse = model_mse.predict(X)
print(f"MSE model coefficient: {model_mse.coef_[0]:.3f}")
print(f"MSE: {mean_squared_error(y, pred_mse):.2f}")

# MAE minimizes to the median → robust to outliers
model_mae = HuberRegressor().fit(X, y)
pred_mae = model_mae.predict(X)
print(f"\nMAE model coefficient: {model_mae.coef_[0]:.3f}")
print(f"MAE: {mean_absolute_error(y, pred_mae):.2f}")

# The difference: MSE is pulled by outliers, MAE is not
print(f"\nTrue coefficient: 3.0")
print(f"MSE is off by: {abs(model_mse.coef_[0] - 3):.3f}")
print(f"MAE is off by: {abs(model_mae.coef_[0] - 3):.3f}")

Key Takeaways